{
  "abstract": "We present short proofs for the main parts of the recently completed solution of Erdős Problem 506, which asks for the minimum number of distinct circles determined by a finite planar point set that is neither collinear nor concyclic. An inversion argument and a seven-point line lemma show that eight points determine at least $17$ circles, and an explicit configuration attains this bound. A single argument combining Melchior's inequality with Langer's incidence bound establishes the sharp formula for every $n\\ge15$. The proofs give concise alternatives to longer arguments in the recent literature; the remaining finite cases are cited separately.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18918",
  "author_total": 3,
  "authors": [
    {
      "name": "Stijn Cambie"
    },
    {
      "name": "Denis Paliy"
    },
    {
      "name": "Rafał Wrona"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3emv3cfc607f6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18918",
  "fulltext_url": null,
  "id": "3emv3cfc607f6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18918",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.HO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7221897616",
  "pdf_url": "https://arxiv.org/pdf/2610.18918",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3emv3cfc607f6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.18918",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18918"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Short Proofs for the Minimum Number of Determined Circles",
  "title_text": "Short Proofs for the Minimum Number of Determined Circles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/3emv3cfc607f6",
  "venue": "arXiv",
  "versions": []
}